Steady vs unsteady flow over cylinder

Aim:

• To simulate flow over cylinder and observe phenomena of Von Karman Vortex.
• Compare steady and unsteady flow over cylinder.

Governing Equations:

The equations given below are solved in solution domain to capture the flow physics. The equations are for 3D geometry, but here we will model 2D geometry for study.

Continuity: `∂ρ/∂t+(∂(ρu))/∂x+(∂(ρv))/∂y+(∂(ρw))/∂z=0`

X-momentum: `(∂(ρu))/∂t+(∂(ρuu))/∂x+(∂(ρuv))/∂y+(∂(ρuw))/∂z=-∂P/∂x+(∂τ_xx)/∂x+(∂τ_xy)/∂y+(∂τ_xz)/∂z+ρg_x`

Y-momentum: `(∂(ρv))/∂t+(∂(ρvu))/∂x+(∂(ρvv))/∂y+(∂(ρvw))/∂z=-∂P/∂y+(∂τ_yx)/∂x+(∂τ_yy)/∂y+(∂τ_yz)/∂z+ρg_y`

Z-momentum: `(∂(ρw))/∂t+(∂(ρwu))/∂x+(∂(ρwv))/∂y+(∂(ρww))/∂z=-∂P/∂z+(∂τ_zx)/∂x+(∂τ_zy)/∂y+(∂τ_zz)/∂z+ρg_z`

While simulating flow over cylinder vortex shedding will appear in flow field. Vortex shedding is responsible for vibrations in body, if it is not properly designed then it may lead to damage to body. There are number of parameters which on vortex shedding depends flow parameters like velocity and type of fluid and geometrical parameters like shape and size of object.

In this study steady and unsteady state simulations are performed for flow over cylinder (Re =100) and vortex shedding effect is observed for both cases.

Some dimensionless numbers required for this study are:

Reynolds Number:

`Re=ρVD/μ`

Where,

- Density of fluid

V- Velocity of fluid

D- Characteristic length of object

- Dynamic viscosity of object

Strouhal’s Number: it is dimensionless number used to detect vortex shedding.

`S= fD/u`

Where,

f- Frequency of shedding

D- Characteristic length of object

u- Velocity of fluid

Work Flow of Project

Problem Definition:

Diameter of Cylinder: 2 m

Upstream distance of cylinder: 20 m

Downstream distance of cylinder: 40 m

Reynolds number of flow: 100

Geometry:

the geometry used for simulation is as shown in above figure.

Meshing:

Meshing is needed to divide computational domain into number of small volumes on which we will solve the above stated governing equations. Sufficiently fine mesh is generated to capture the rapid change in variables i.e. velocity, pressure etc. due to vortex shedding. An unstructured triangular mesh is used.

As you move towards the wall of cylinder from freestream, the velocity decreases non-linearly up to zero velocity at wall due to no-slip condition. And we can observe large gradient in velocity near cylinder wall. Hence, in order to capture this large gradient correctly an inflation layers are added near wall of cylinder with first layer thickness 1e-3 m.

Mesh Details:

Element type: Triangular

Element size: 0.25 m

Number of inflation layers: 6

First layer thickness: 5e-3 m

Growth size: 1.2

Mesh quality: as shown in figure below.

Setting and Results:

User defined fluid is defined in Materials section. With parameters as:

Density of fluid: 1 kg/m³.

Viscosity: 0.02 N-s/m².

This parameters will give Reynolds number as 100. Hence, viscous model is selected as laminar.

• Steady State Analysis:

ANSYS Fluent solver is used for simulation. The double precision solver setting with pressure based solver and steady state analysis is chosen. And continuity and momentum equations are used for solution.

Boundary conditions:

Inlet: Velocity Inlet, V= 1 m/s

Outlet: Pressure Outet, Guage pressure =0 Pa

Operating Pressure: 1Bar

Wall: No-slip condition.

Residuals:

The graph shown above shows the residuals which are used to determine convergence is achieved or not. As we can see from figure shown above the solution is converged around 300 iterations.

And one vertex is also created to monitor the vortex formation. It is observed that the vortex reaches steady state after 350 iterations.

Velocity and Pressure Contour plots:

The figure below shows contour plots for velocity and pressure. We can observe the vortex shedding on downstream of cylinder.

Strouhal Number Calculation:

In steady state simulation the solution is not monitored with time and it fives only output results. Hence it is impossible to calculate the cortex shedding frequency. Hence strouhal number cannot be calculated using this solution.

Transient Simulation:

The fluid material and boundary conditions are used as it is from steady state analysis, with transient parameters

Time step: 0.1 sec

Number of iterations per time step: 20

Number of time steps: 1800

Residuals:

From the figures shown above it is observed that the solution is converged.

Velocity and Pressure Contour plots:

The figure below shows contour plots for velocity and pressure. We can observe the vortex shedding on downstream of cylinder.

Conclusion:

The vortex shedding phenomenon is observed for laminar flow over cylinder for steady and unsteady cases.

When we forced solution to be steady (Reynolds number 50), whereas in actual case the flow over cylinder is unsteady Karman Vortex Street is not formed downstream of cylinder.

Poor grid resolution will give incorrect solution, particularly with boundary layer separation. Continued refinement of grid will not give physically accurate solution if boundary conditions are not assigned properly.

and also we can see from velocity contour plots that transient simulation predicts better Karman Voretx Street than steady simulation.